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Aliquot sequence - Wikipedia

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href="#Catalan–Dickson_conjecture"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Catalan–Dickson conjecture</span> </div> </a> <ul id="toc-Catalan–Dickson_conjecture-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Systematically_searching_for_aliquot_sequences" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Systematically_searching_for_aliquot_sequences"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Systematically searching for aliquot sequences</span> </div> </a> <ul id="toc-Systematically_searching_for_aliquot_sequences-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> 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Available in 14 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-14" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">14 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AA%D8%AA%D8%A7%D9%84%D9%8A%D8%A9_%D8%AA%D8%AC%D8%B2%D9%8A%D8%A6%D9%8A%D8%A9" title="متتالية تجزيئية – Arabic" lang="ar" hreflang="ar" data-title="متتالية تجزيئية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Alikvotf%C3%B8lge" title="Alikvotfølge – Danish" lang="da" hreflang="da" data-title="Alikvotfølge" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Inhaltskette" title="Inhaltskette – German" lang="de" hreflang="de" data-title="Inhaltskette" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Sucesi%C3%B3n_al%C3%ADcuota" title="Sucesión alícuota – Spanish" lang="es" hreflang="es" data-title="Sucesión alícuota" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Suite_aliquote" title="Suite aliquote – French" lang="fr" hreflang="fr" data-title="Suite aliquote" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A1%D7%93%D7%A8%D7%AA_%D7%9E%D7%97%D7%9C%D7%A7%D7%99%D7%9D" title="סדרת מחלקים – Hebrew" lang="he" hreflang="he" data-title="סדרת מחלקים" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Oszt%C3%B3%C3%B6sszeg-sorozat" title="Osztóösszeg-sorozat – Hungarian" lang="hu" hreflang="hu" data-title="Osztóösszeg-sorozat" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Aliquotrij" title="Aliquotrij – Dutch" lang="nl" hreflang="nl" data-title="Aliquotrij" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%A2%E3%83%AA%E3%82%B3%E3%83%83%E3%83%88%E6%95%B0%E5%88%97" title="アリコット数列 – Japanese" lang="ja" hreflang="ja" data-title="アリコット数列" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Serie_alicot%C4%83" title="Serie alicotă – Romanian" lang="ro" hreflang="ro" data-title="Serie alicotă" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D0%BB%D0%B8%D0%BA%D0%B2%D0%BE%D1%82%D0%BD%D0%B0%D1%8F_%D0%BF%D0%BE%D1%81%D0%BB%D0%B5%D0%B4%D0%BE%D0%B2%D0%B0%D1%82%D0%B5%D0%BB%D1%8C%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Аликвотная последовательность – Russian" lang="ru" hreflang="ru" data-title="Аликвотная последовательность" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Alikvotno_zaporedje" title="Alikvotno zaporedje – Slovenian" lang="sl" hreflang="sl" data-title="Alikvotno zaporedje" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link 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banned users from editing it"><img alt="Page semi-protected" src="//upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/20px-Semi-protection-shackle.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/30px-Semi-protection-shackle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/40px-Semi-protection-shackle.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Mathematical recursive sequence</div> <p> <style data-mw-deduplicate="TemplateStyles:r1233989161">.mw-parser-output .unsolved{margin:0.5em 0 1em 1em;border:#ccc solid;padding:0.35em 0.35em 0.35em 2.2em;background-color:var(--background-color-interactive-subtle);background-image:url("https://upload.wikimedia.org/wikipedia/commons/2/26/Question%2C_Web_Fundamentals.svg");background-position:top 50%left 0.35em;background-size:1.5em;background-repeat:no-repeat}@media(min-width:720px){.mw-parser-output .unsolved{clear:right;float:right;max-width:25%}}.mw-parser-output .unsolved-label{font-weight:bold}.mw-parser-output .unsolved-body{margin:0.35em;font-style:italic}.mw-parser-output .unsolved-more{font-size:smaller}</style> </p> <div role="note" aria-labelledby="unsolved-label-mathematics" class="unsolved"> <div><span class="unsolved-label" id="unsolved-label-mathematics">Unsolved problem in mathematics</span>:</div> <div class="unsolved-body">Do all aliquot sequences eventually end with a prime number, a perfect number, or a set of amicable or sociable numbers? (Catalan's aliquot sequence conjecture)</div> <div class="unsolved-more"><a href="/wiki/List_of_unsolved_problems_in_mathematics" title="List of unsolved problems in mathematics">(more unsolved problems in mathematics)</a></div> </div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, an <b>aliquot sequence</b> is a sequence of positive integers in which each term is the sum of the <a href="/wiki/Proper_divisor" class="mw-redirect" title="Proper divisor">proper divisors</a> of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition_and_overview">Definition and overview</h2></div> <p>The <a href="https://en.wiktionary.org/wiki/aliquot" class="extiw" title="wiktionary:aliquot">aliquot</a> sequence starting with a positive integer <span class="texhtml mvar" style="font-style:italic;">k</span> can be defined formally in terms of the <a href="/wiki/Divisor_function" title="Divisor function">sum-of-divisors function</a> <span class="texhtml"><i>σ</i><sub>1</sub></span> or the <a href="/wiki/Aliquot_sum" title="Aliquot sum">aliquot sum</a> function <span class="texhtml mvar" style="font-style:italic;">s</span> in the following way:<sup id="cite_ref-mw_1-0" class="reference"><a href="#cite_note-mw-1"><span class="cite-bracket">&#91;</span>1<span class="cite-bracket">&#93;</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}s_{0}&amp;=k\\[4pt]s_{n}&amp;=s(s_{n-1})=\sigma _{1}(s_{n-1})-s_{n-1}\quad {\text{if}}\quad s_{n-1}&gt;0\\[4pt]s_{n}&amp;=0\quad {\text{if}}\quad s_{n-1}=0\\[4pt]s(0)&amp;={\text{undefined}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.7em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>k</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>s</mi> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>&#x2212;<!-- − --></mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>&#x03C3;<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>&#x2212;<!-- − --></mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>&#x2212;<!-- − --></mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>&#x2212;<!-- − --></mo> <mn>1</mn> </mrow> </msub> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>if</mtext> </mrow> <mspace width="1em" /> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>&#x2212;<!-- − --></mo> <mn>1</mn> </mrow> </msub> <mo>&gt;</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>if</mtext> </mrow> <mspace width="1em" /> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>&#x2212;<!-- − --></mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>s</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>undefined</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s_{0}&amp;=k\\[4pt]s_{n}&amp;=s(s_{n-1})=\sigma _{1}(s_{n-1})-s_{n-1}\quad {\text{if}}\quad s_{n-1}&gt;0\\[4pt]s_{n}&amp;=0\quad {\text{if}}\quad s_{n-1}=0\\[4pt]s(0)&amp;={\text{undefined}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10966fe8df0b3042041fe3e8f5171ecf894eac8f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:48.844ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}s_{0}&amp;=k\\[4pt]s_{n}&amp;=s(s_{n-1})=\sigma _{1}(s_{n-1})-s_{n-1}\quad {\text{if}}\quad s_{n-1}&gt;0\\[4pt]s_{n}&amp;=0\quad {\text{if}}\quad s_{n-1}=0\\[4pt]s(0)&amp;={\text{undefined}}\end{aligned}}}"></span> If the <span class="texhtml"><i>s</i><sub><i>n</i>-1</sub> = 0</span> condition is added, then the terms after 0 are all 0, and all aliquot sequences would be infinite, and we can conjecture that all aliquot sequences are <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">convergent</a>, the limit of these sequences are usually 0 or 6. </p><p>For example, the aliquot sequence of 10 is <span class="nowrap">10, 8, 7, 1, 0</span> because: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sigma _{1}(10)-10&amp;=5+2+1=8,\\[4pt]\sigma _{1}(8)-8&amp;=4+2+1=7,\\[4pt]\sigma _{1}(7)-7&amp;=1,\\[4pt]\sigma _{1}(1)-1&amp;=0.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.7em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>&#x03C3;<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>10</mn> <mo stretchy="false">)</mo> <mo>&#x2212;<!-- − --></mo> <mn>10</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>5</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>8</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&#x03C3;<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>8</mn> <mo stretchy="false">)</mo> <mo>&#x2212;<!-- − --></mo> <mn>8</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>4</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>7</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&#x03C3;<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mo>&#x2212;<!-- − --></mo> <mn>7</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&#x03C3;<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>&#x2212;<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sigma _{1}(10)-10&amp;=5+2+1=8,\\[4pt]\sigma _{1}(8)-8&amp;=4+2+1=7,\\[4pt]\sigma _{1}(7)-7&amp;=1,\\[4pt]\sigma _{1}(1)-1&amp;=0.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da312fbc77fa014a3288a960a0df3cbfac46d534" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.005ex; width:29.607ex; height:15.176ex;" alt="{\displaystyle {\begin{aligned}\sigma _{1}(10)-10&amp;=5+2+1=8,\\[4pt]\sigma _{1}(8)-8&amp;=4+2+1=7,\\[4pt]\sigma _{1}(7)-7&amp;=1,\\[4pt]\sigma _{1}(1)-1&amp;=0.\end{aligned}}}"></span> </p><p>Many aliquot sequences terminate at zero; all such sequences necessarily end with a <a href="/wiki/Prime_number" title="Prime number">prime number</a> followed by 1 (since the only proper divisor of a prime is 1), followed by 0 (since 1 has no proper divisors). See (sequence <span class="nowrap external"><a href="//oeis.org/A080907" class="extiw" title="oeis:A080907">A080907</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) for a list of such numbers up to 75. There are a variety of ways in which an aliquot sequence might not terminate: </p> <ul><li>A <a href="/wiki/Perfect_number" title="Perfect number">perfect number</a> has a repeating aliquot sequence of period 1. The aliquot sequence of 6, for example, is <span class="nowrap">6, 6, 6, 6, ...</span></li> <li>An <a href="/wiki/Amicable_number" class="mw-redirect" title="Amicable number">amicable number</a> has a repeating aliquot sequence of period 2. For instance, the aliquot sequence of 220 is <span class="nowrap">220, 284, 220, 284, ...</span></li> <li>A <a href="/wiki/Sociable_number" title="Sociable number">sociable number</a> has a repeating aliquot sequence of period 3 or greater. (Sometimes the term <i>sociable number</i> is used to encompass amicable numbers as well.) For instance, the aliquot sequence of 1264460 is <span class="nowrap">1264460, 1547860, 1727636, 1305184, 1264460, ...</span></li> <li>Some numbers have an aliquot sequence which is eventually periodic, but the number itself is not perfect, amicable, or sociable. For instance, the aliquot sequence of 95 is <span class="nowrap">95, 25, 6, 6, 6, 6, ...</span> Numbers like 95 that are not perfect, but have an eventually repeating aliquot sequence of period 1 are called <b>aspiring numbers</b>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">&#91;</span>2<span class="cite-bracket">&#93;</span></a></sup></li></ul> <table class="wikitable mw-collapsible mw-collapsed"> <caption class="nowrap">Aliquot sequences from 0 to 47 </caption> <tbody><tr> <th><span class="texhtml mvar" style="font-style:italic;">n</span></th> <th>Aliquot sequence of <span class="texhtml mvar" style="font-style:italic;">n</span></th> <th>Length (<span class="nowrap external"><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>:&#160;<a href="//oeis.org/A098007" class="extiw" title="oeis:A098007">A098007</a></span>) </th></tr> <tr> <th>0 </th> <td>0</td> <td>1 </td></tr> <tr> <th>1 </th> <td>1, 0</td> <td>2 </td></tr> <tr> <th>2 </th> <td>2, 1, 0</td> <td>3 </td></tr> <tr> <th>3 </th> <td>3, 1, 0</td> <td>3 </td></tr> <tr> <th>4 </th> <td>4, 3, 1, 0</td> <td>4 </td></tr> <tr> <th>5 </th> <td>5, 1, 0</td> <td>3 </td></tr> <tr> <th>6 </th> <td>6</td> <td>1 </td></tr> <tr> <th>7 </th> <td>7, 1, 0</td> <td>3 </td></tr> <tr> <th>8 </th> <td>8, 7, 1, 0</td> <td>4 </td></tr> <tr> <th>9 </th> <td>9, 4, 3, 1, 0</td> <td>5 </td></tr> <tr> <th>10 </th> <td>10, 8, 7, 1, 0</td> <td>5 </td></tr> <tr> <th>11 </th> <td>11, 1, 0</td> <td>3 </td></tr> <tr> <th>12 </th> <td>12, 16, 15, 9, 4, 3, 1, 0</td> <td>8 </td></tr> <tr> <th>13 </th> <td>13, 1, 0</td> <td>3 </td></tr> <tr> <th>14 </th> <td>14, 10, 8, 7, 1, 0</td> <td>6 </td></tr> <tr> <th>15 </th> <td>15, 9, 4, 3, 1, 0</td> <td>6 </td></tr> <tr> <th>16 </th> <td>16, 15, 9, 4, 3, 1, 0</td> <td>7 </td></tr> <tr> <th>17 </th> <td>17, 1, 0</td> <td>3 </td></tr> <tr> <th>18 </th> <td>18, 21, 11, 1, 0</td> <td>5 </td></tr> <tr> <th>19 </th> <td>19, 1, 0</td> <td>3 </td></tr> <tr> <th>20 </th> <td>20, 22, 14, 10, 8, 7, 1, 0</td> <td>8 </td></tr> <tr> <th>21 </th> <td>21, 11, 1, 0</td> <td>4 </td></tr> <tr> <th>22 </th> <td>22, 14, 10, 8, 7, 1, 0</td> <td>7 </td></tr> <tr> <th>23 </th> <td>23, 1, 0</td> <td>3 </td></tr> <tr> <th>24 </th> <td>24, 36, 55, 17, 1, 0</td> <td>6 </td></tr> <tr> <th>25 </th> <td>25, 6</td> <td>2 </td></tr> <tr> <th>26 </th> <td>26, 16, 15, 9, 4, 3, 1, 0</td> <td>8 </td></tr> <tr> <th>27 </th> <td>27, 13, 1, 0</td> <td>4 </td></tr> <tr> <th>28 </th> <td>28</td> <td>1 </td></tr> <tr> <th>29 </th> <td>29, 1, 0</td> <td>3 </td></tr> <tr> <th>30 </th> <td>30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0</td> <td>16 </td></tr> <tr> <th>31 </th> <td>31, 1, 0</td> <td>3 </td></tr> <tr> <th>32 </th> <td>32, 31, 1, 0</td> <td>4 </td></tr> <tr> <th>33 </th> <td>33, 15, 9, 4, 3, 1, 0</td> <td>7 </td></tr> <tr> <th>34 </th> <td>34, 20, 22, 14, 10, 8, 7, 1, 0</td> <td>9 </td></tr> <tr> <th>35 </th> <td>35, 13, 1, 0</td> <td>4 </td></tr> <tr> <th>36 </th> <td>36, 55, 17, 1, 0</td> <td>5 </td></tr> <tr> <th>37 </th> <td>37, 1, 0</td> <td>3 </td></tr> <tr> <th>38 </th> <td>38, 22, 14, 10, 8, 7, 1, 0</td> <td>8 </td></tr> <tr> <th>39 </th> <td>39, 17, 1, 0</td> <td>4 </td></tr> <tr> <th>40 </th> <td>40, 50, 43, 1, 0</td> <td>5 </td></tr> <tr> <th>41 </th> <td>41, 1, 0</td> <td>3 </td></tr> <tr> <th>42 </th> <td>42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0</td> <td>15 </td></tr> <tr> <th>43 </th> <td>43, 1, 0</td> <td>3 </td></tr> <tr> <th>44 </th> <td>44, 40, 50, 43, 1, 0</td> <td>6 </td></tr> <tr> <th>45 </th> <td>45, 33, 15, 9, 4, 3, 1, 0</td> <td>8 </td></tr> <tr> <th>46 </th> <td>46, 26, 16, 15, 9, 4, 3, 1, 0</td> <td>9 </td></tr> <tr> <th>47 </th> <td>47, 1, 0</td> <td>3 </td></tr></tbody></table> <p>The lengths of the aliquot sequences that start at <span class="texhtml mvar" style="font-style:italic;">n</span> are </p> <dl><dd>1, 2, 2, 3, 2, 1, 2, 3, 4, 4, 2, 7, 2, 5, 5, 6, 2, 4, 2, 7, 3, 6, 2, 5, 1, 7, 3, 1, 2, 15, 2, 3, 6, 8, 3, 4, 2, 7, 3, 4, 2, 14, 2, 5, 7, 8, 2, 6, 4, 3, ... (sequence <span class="nowrap external"><a href="//oeis.org/A044050" class="extiw" title="oeis:A044050">A044050</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> <p>The final terms (excluding 1) of the aliquot sequences that start at <span class="texhtml mvar" style="font-style:italic;">n</span> are </p> <dl><dd>1, 2, 3, 3, 5, 6, 7, 7, 3, 7, 11, 3, 13, 7, 3, 3, 17, 11, 19, 7, 11, 7, 23, 17, 6, 3, 13, 28, 29, 3, 31, 31, 3, 7, 13, 17, 37, 7, 17, 43, 41, 3, 43, 43, 3, 3, 47, 41, 7, 43, ... (sequence <span class="nowrap external"><a href="//oeis.org/A115350" class="extiw" title="oeis:A115350">A115350</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> <p>Numbers whose aliquot sequence terminates in 1 are </p> <dl><dd>1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, ... (sequence <span class="nowrap external"><a href="//oeis.org/A080907" class="extiw" title="oeis:A080907">A080907</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> <p>Numbers whose aliquot sequence known to terminate in a <a href="/wiki/Perfect_number" title="Perfect number">perfect number</a>, other than perfect numbers themselves (6, 28, 496, ...), are </p> <dl><dd>25, 95, 119, 143, 417, 445, 565, 608, 650, 652, 675, 685, 783, 790, 909, 913, ... (sequence <span class="nowrap external"><a href="//oeis.org/A063769" class="extiw" title="oeis:A063769">A063769</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> <p>Numbers whose aliquot sequence terminates in a cycle with length at least 2 are </p> <dl><dd>220, 284, 562, 1064, 1184, 1188, 1210, 1308, 1336, 1380, 1420, 1490, 1604, 1690, 1692, 1772, 1816, 1898, 2008, 2122, 2152, 2172, 2362, ... (sequence <span class="nowrap external"><a href="//oeis.org/A121507" class="extiw" title="oeis:A121507">A121507</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> <p>Numbers whose aliquot sequence is not known to be finite or eventually periodic are </p> <dl><dd>276, 306, 396, 552, 564, 660, 696, 780, 828, 888, 966, 996, 1074, 1086, 1098, 1104, 1134, 1218, 1302, 1314, 1320, 1338, 1350, 1356, 1392, 1398, 1410, 1464, 1476, 1488, ... (sequence <span class="nowrap external"><a href="//oeis.org/A131884" class="extiw" title="oeis:A131884">A131884</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> <p>A number that is never the successor in an aliquot sequence is called an <a href="/wiki/Untouchable_number" title="Untouchable number">untouchable number</a>. </p> <dl><dd><a href="/wiki/2_(number)" class="mw-redirect" title="2 (number)">2</a>, <a href="/wiki/5_(number)" class="mw-redirect" title="5 (number)">5</a>, <a href="/wiki/52_(number)" title="52 (number)">52</a>, <a href="/wiki/88_(number)" title="88 (number)">88</a>, <a href="/wiki/96_(number)" title="96 (number)">96</a>, <a href="/wiki/120_(number)" title="120 (number)">120</a>, <a href="/wiki/124_(number)" title="124 (number)">124</a>, <a href="/wiki/146_(number)" title="146 (number)">146</a>, <a href="/wiki/162_(number)" title="162 (number)">162</a>, <a href="/wiki/188_(number)" title="188 (number)">188</a>, <a href="/wiki/206_(number)" title="206 (number)">206</a>, <a href="/wiki/210_(number)" title="210 (number)">210</a>, <a href="/wiki/216_(number)" title="216 (number)">216</a>, <a href="/wiki/238_(number)" title="238 (number)">238</a>, <a href="/wiki/246_(number)" title="246 (number)">246</a>, <a href="/wiki/248_(number)" title="248 (number)">248</a>, 262, 268, <a href="/wiki/276_(number)" title="276 (number)">276</a>, <a href="/wiki/288_(number)" title="288 (number)">288</a>, <a href="/wiki/290_(number)" title="290 (number)">290</a>, 292, 304, 306, 322, 324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, ... (sequence <span class="nowrap external"><a href="//oeis.org/A005114" class="extiw" title="oeis:A005114">A005114</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Catalan–Dickson_conjecture"><span id="Catalan.E2.80.93Dickson_conjecture"></span>Catalan–Dickson conjecture</h2></div> <p>An important <a href="/wiki/Conjecture" title="Conjecture">conjecture</a> due to <a href="/wiki/Eug%C3%A8ne_Charles_Catalan" title="Eugène Charles Catalan">Catalan</a>, sometimes called the Catalan–<a href="/wiki/Leonard_Eugene_Dickson" title="Leonard Eugene Dickson">Dickson</a> conjecture, is that every aliquot sequence ends in one of the above ways: with a prime number, a perfect number, or a set of amicable or sociable numbers.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">&#91;</span>3<span class="cite-bracket">&#93;</span></a></sup> The alternative would be that a number exists whose aliquot sequence is infinite yet never repeats. Any one of the many numbers whose aliquot sequences have not been fully determined might be such a number. The first five candidate numbers are often called the <b>Lehmer five</b> (named after <a href="/wiki/Derrick_Henry_Lehmer" class="mw-redirect" title="Derrick Henry Lehmer">D.H. Lehmer</a>): <a href="/wiki/276_(number)" title="276 (number)">276</a>, 552, 564, 660, and 966.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">&#91;</span>4<span class="cite-bracket">&#93;</span></a></sup> However, it is worth noting that 276 may reach a high apex in its aliquot sequence and then descend; the number 138 reaches a peak of 179931895322 before returning to 1. </p><p><a href="/wiki/Richard_K._Guy" title="Richard K. Guy">Guy</a> and <a href="/wiki/John_Selfridge" title="John Selfridge">Selfridge</a> believe the Catalan–Dickson conjecture is false (so they conjecture some aliquot sequences are <a href="/wiki/Bounded_function" title="Bounded function">unbounded</a> above (i.e., diverge)).<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">&#91;</span>5<span class="cite-bracket">&#93;</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Systematically_searching_for_aliquot_sequences">Systematically searching for aliquot sequences</h2></div> <p>The aliquot sequence can be represented as a <a href="/wiki/Directed_graph" title="Directed graph">directed graph</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{n,s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{n,s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feb647f8d64df5cc70abfd55ff8b9029b5415b65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.274ex; height:2.843ex;" alt="{\displaystyle G_{n,s}}"></span>, for a given integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d588236e5ce92331acd2415671cbab71e90cd985" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.111ex; height:2.843ex;" alt="{\displaystyle s(k)}"></span> denotes the sum of the proper divisors of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">&#91;</span>6<span class="cite-bracket">&#93;</span></a></sup> <a href="/wiki/Cycle_(graph_theory)" title="Cycle (graph theory)">Cycles</a> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{n,s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{n,s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feb647f8d64df5cc70abfd55ff8b9029b5415b65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.274ex; height:2.843ex;" alt="{\displaystyle G_{n,s}}"></span> represent sociable numbers within the interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [1,n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [1,n]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c79af450e22e8fd23f28e6be4cb23a47b24c1ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.885ex; height:2.843ex;" alt="{\displaystyle [1,n]}"></span>. Two special cases are loops that represent <a href="/wiki/Perfect_numbers" class="mw-redirect" title="Perfect numbers">perfect numbers</a> and cycles of length two that represent <a href="/wiki/Amicable_pairs" class="mw-redirect" title="Amicable pairs">amicable pairs</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <ul><li><a href="/wiki/Arithmetic_dynamics" title="Arithmetic dynamics">Arithmetic dynamics</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-mw-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-mw_1-0">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Aliquot_Sequence"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/AliquotSequence.html">"Aliquot Sequence"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=unknown&amp;rft.jtitle=MathWorld&amp;rft.atitle=Aliquot+Sequence&amp;rft.au=Weisstein%2C+Eric+W.&amp;rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FAliquotSequence.html&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AAliquot+sequence" class="Z3988"></span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSloane_&quot;A063769&quot;" class="citation web cs1"><a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane, N.&#160;J.&#160;A.</a> (ed.). <a rel="nofollow" class="external text" href="https://oeis.org/A063769">"Sequence&#x20;A063769&#x20;(Aspiring numbers: numbers whose aliquot sequence terminates in a perfect number.)"</a>. <i>The <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a></i>. OEIS Foundation.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=unknown&amp;rft.jtitle=The+On-Line+Encyclopedia+of+Integer+Sequences&amp;rft.atitle=Sequence%26%23x20%3BA063769%26%23x20%3B%28Aspiring+numbers%3A+numbers+whose+aliquot+sequence+terminates+in+a+perfect+number.%29&amp;rft_id=https%3A%2F%2Foeis.org%2FA063769&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AAliquot+sequence" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Catalan&#39;s_Aliquot_Sequence_Conjecture"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. 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Retrieved <span class="nowrap">June 14,</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=unknown&amp;rft.btitle=Lehmer+Five&amp;rft.date=2014-05-24&amp;rft.aulast=Creyaufm%C3%BCller&amp;rft.aufirst=Wolfgang&amp;rft_id=http%3A%2F%2Fwww.aliquot.de%2Flehmer.htm&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AAliquot+sequence" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">A. S. Mosunov, <a rel="nofollow" class="external text" href="http://www.cs.uleth.ca/~hadi/2016-09-29-aliquot_sequences.pdf">What do we know about aliquot sequences?</a></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRochaThatte2015" class="citation cs2">Rocha, Rodrigo Caetano; Thatte, Bhalchandra (2015), <i>Distributed cycle detection in large-scale sparse graphs</i>, Simpósio Brasileiro de Pesquisa Operacional (SBPO), <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.13140%2FRG.2.1.1233.8640">10.13140/RG.2.1.1233.8640</a></cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Distributed+cycle+detection+in+large-scale+sparse+graphs&amp;rft.pub=Simp%C3%B3sio+Brasileiro+de+Pesquisa+Operacional+%28SBPO%29&amp;rft.date=2015&amp;rft_id=info%3Adoi%2F10.13140%2FRG.2.1.1233.8640&amp;rft.aulast=Rocha&amp;rft.aufirst=Rodrigo+Caetano&amp;rft.au=Thatte%2C+Bhalchandra&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AAliquot+sequence" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li>Manuel Benito; Wolfgang Creyaufmüller; Juan Luis Varona; Paul Zimmermann. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20041015194432/http://www.expmath.org/expmath/volumes/11/11.2/3630finishes1.pdf"><i>Aliquot Sequence 3630 Ends After Reaching 100 Digits</i></a>. Experimental Mathematics, vol. 11, num. 2, Natick, MA, 2002, p.&#160;201–206.</li> <li>W. Creyaufmüller. <i>Primzahlfamilien - Das Catalan'sche Problem und die Familien der Primzahlen im Bereich 1 bis 3000 im Detail</i>. Stuttgart 2000 (3rd ed.), 327p.</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div> <ul><li><a rel="nofollow" class="external text" href="http://www.rechenkraft.net/aliquot/AllSeq.html">Current status of aliquot sequences with start term below 2 million</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20140502102524/http://amicable.homepage.dk/tables.htm">Tables of Aliquot Cycles</a> (J.O.M. Pedersen)</li> <li><a rel="nofollow" class="external text" href="http://www.aliquot.de/aliquote.htm">Aliquot Page</a> (Wolfgang Creyaufmüller)</li> <li><a rel="nofollow" class="external text" href="http://christophe.clavier.free.fr/Aliquot/site/Aliquot.html">Aliquot sequences</a> (Christophe Clavier)</li> <li><a rel="nofollow" class="external text" href="http://www.mersenneforum.org/forumdisplay.php?f=90">Forum on calculating aliquot sequences</a> (MersenneForum)</li> <li><a rel="nofollow" class="external text" href="http://www.rieselprime.de/Others/Aliquot000.htm">Aliquot sequence summary page for sequences up to 100000 (there are similar pages for higher ranges)</a> (Karsten Bonath)</li> <li><a rel="nofollow" class="external text" href="http://www.aliquotes.com">Active research site on aliquot sequences</a> (Jean-Luc Garambois) <span class="languageicon">(in French)</span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output 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.navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Divisor_classes" title="Template:Divisor classes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Divisor_classes" title="Template talk:Divisor classes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Divisor_classes" title="Special:EditPage/Template:Divisor classes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Divisibility-based_sets_of_integers" style="font-size:114%;margin:0 4em">Divisibility-based sets of integers</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Overview</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Integer_factorization" title="Integer factorization">Integer factorization</a></li> <li><a href="/wiki/Divisor" title="Divisor">Divisor</a></li> <li><a href="/wiki/Unitary_divisor" title="Unitary divisor">Unitary divisor</a></li> <li><a href="/wiki/Divisor_function" title="Divisor function">Divisor function</a></li> <li><a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">Prime factor</a></li> <li><a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">Fundamental theorem of arithmetic</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="7" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Lattice_of_the_divisibility_of_60.svg" class="mw-file-description" title="Divisibility of 60"><img alt="Divisibility of 60" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/175px-Lattice_of_the_divisibility_of_60.svg.png" decoding="async" width="175" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/263px-Lattice_of_the_divisibility_of_60.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/350px-Lattice_of_the_divisibility_of_60.svg.png 2x" data-file-width="313" data-file-height="250" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Factorization forms</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Prime_number" title="Prime number">Prime</a></li> <li><a href="/wiki/Composite_number" title="Composite number">Composite</a></li> <li><a href="/wiki/Semiprime" title="Semiprime">Semiprime</a></li> <li><a href="/wiki/Pronic_number" title="Pronic number">Pronic</a></li> <li><a href="/wiki/Sphenic_number" title="Sphenic number">Sphenic</a></li> <li><a href="/wiki/Square-free_integer" title="Square-free integer">Square-free</a></li> <li><a href="/wiki/Powerful_number" title="Powerful number">Powerful</a></li> <li><a href="/wiki/Perfect_power" title="Perfect power">Perfect power</a></li> <li><a href="/wiki/Achilles_number" title="Achilles number">Achilles</a></li> <li><a href="/wiki/Smooth_number" title="Smooth number">Smooth</a></li> <li><a href="/wiki/Regular_number" title="Regular number">Regular</a></li> <li><a href="/wiki/Rough_number" title="Rough number">Rough</a></li> <li><a href="/wiki/Unusual_number" title="Unusual number">Unusual</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constrained divisor sums</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Perfect_number" title="Perfect number">Perfect</a></li> <li><a href="/wiki/Almost_perfect_number" title="Almost perfect number">Almost perfect</a></li> <li><a href="/wiki/Quasiperfect_number" title="Quasiperfect number">Quasiperfect</a></li> <li><a href="/wiki/Multiply_perfect_number" title="Multiply perfect number">Multiply perfect</a></li> <li><a href="/wiki/Hemiperfect_number" title="Hemiperfect number">Hemiperfect</a></li> <li><a href="/wiki/Hyperperfect_number" title="Hyperperfect number">Hyperperfect</a></li> <li><a href="/wiki/Superperfect_number" title="Superperfect number">Superperfect</a></li> <li><a href="/wiki/Unitary_perfect_number" title="Unitary perfect number">Unitary perfect</a></li> <li><a href="/wiki/Semiperfect_number" title="Semiperfect number">Semiperfect</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Descartes_number" title="Descartes number">Descartes</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Nicolas_number" title="Erdős–Nicolas number">Erdős–Nicolas</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">With many divisors</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abundant_number" title="Abundant number">Abundant</a></li> <li><a href="/wiki/Primitive_abundant_number" title="Primitive abundant number">Primitive abundant</a></li> <li><a href="/wiki/Highly_abundant_number" title="Highly abundant number">Highly abundant</a></li> <li><a href="/wiki/Superabundant_number" title="Superabundant number">Superabundant</a></li> <li><a href="/wiki/Colossally_abundant_number" title="Colossally abundant number">Colossally abundant</a></li> <li><a href="/wiki/Highly_composite_number" title="Highly composite number">Highly composite</a></li> <li><a href="/wiki/Superior_highly_composite_number" title="Superior highly composite number">Superior highly composite</a></li> <li><a href="/wiki/Weird_number" title="Weird number">Weird</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink selflink">Aliquot sequence</a>-related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Untouchable_number" title="Untouchable number">Untouchable</a></li> <li><a href="/wiki/Amicable_numbers" title="Amicable numbers">Amicable</a> (<a href="/wiki/Amicable_triple" title="Amicable triple">Triple</a>)</li> <li><a href="/wiki/Sociable_number" title="Sociable number">Sociable</a></li> <li><a href="/wiki/Betrothed_numbers" title="Betrothed numbers">Betrothed</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Radix" title="Radix">Base</a>-dependent</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Equidigital_number" title="Equidigital number">Equidigital</a></li> <li><a href="/wiki/Extravagant_number" title="Extravagant number">Extravagant</a></li> <li><a href="/wiki/Frugal_number" title="Frugal number">Frugal</a></li> <li><a href="/wiki/Harshad_number" title="Harshad number">Harshad</a></li> <li><a href="/wiki/Polydivisible_number" title="Polydivisible number">Polydivisible</a></li> <li><a href="/wiki/Smith_number" title="Smith number">Smith</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other sets</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetic_number" title="Arithmetic number">Arithmetic</a></li> <li><a href="/wiki/Deficient_number" title="Deficient number">Deficient</a></li> <li><a href="/wiki/Friendly_number" title="Friendly number">Friendly</a></li> <li><a href="/wiki/Friendly_number#Solitary_numbers" title="Friendly number">Solitary</a></li> <li><a href="/wiki/Sublime_number" title="Sublime number">Sublime</a></li> <li><a href="/wiki/Harmonic_divisor_number" title="Harmonic divisor number">Harmonic divisor</a></li> <li><a href="/wiki/Refactorable_number" title="Refactorable number">Refactorable</a></li> <li><a href="/wiki/Superperfect_number" title="Superperfect number">Superperfect</a></li></ul> </div></td></tr></tbody></table></div> <!-- NewPP limit report Parsed by mw‐web.codfw.main‐7fc47fc68d‐9mwsm Cached time: 20241128182222 Cache expiry: 2592000 Reduced expiry: false Complications: [vary‐revision‐sha1, show‐toc] CPU time usage: 0.451 seconds Real time usage: 0.627 seconds Preprocessor visited node count: 1135/1000000 Post‐expand include size: 28531/2097152 bytes Template argument size: 1588/2097152 bytes Highest expansion depth: 10/100 Expensive parser function count: 3/500 Unstrip recursion depth: 1/20 Unstrip post‐expand size: 27919/5000000 bytes Lua time usage: 0.313/10.000 seconds Lua memory usage: 16392830/52428800 bytes Number of Wikibase entities loaded: 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